Math 2.0

Global analysis, analysis on manifolds Definition

Pfaffian system

Builds on Exterior differential systemOne-formDifferential formSmooth manifoldFrobenius theorem

Definition

Let MM be a smooth manifold of dimension nn. A Pfaffian system on MM is an exterior differential system ℐ\mathcal{I} generated, as a differential ideal, by one-forms θ1,…,θs\theta^1, \dots, \theta^s. So ℐ\mathcal{I} is the algebraic ideal generated by θ1,…,θs\theta^1, \dots, \theta^s and dθ1,…,dθsd\theta^1, \dots, d\theta^s. One usually asks that θ1,…,θs\theta^1, \dots, \theta^s are linearly independent at each point. Then 𝒟={v∈TM:θa(v)=0 for a=1,…,s}\mathcal{D} = \set{ v \in TM : \theta^a(v) = 0 \text{ for } a = 1, \dots, s } is a distribution of rank n−sn - s, and an immersed submanifold is an integral manifold of ℐ\mathcal{I} exactly when it is tangent to 𝒟\mathcal{D}.

Remarks

The system is completely integrable if dθa∧θ1∧⋯∧θs=0d\theta^a \wedge \theta^1 \wedge \dots \wedge \theta^s = 0 for a=1,…,sa = 1, \dots, s, that is, if each dθad\theta^a lies in the algebraic ideal generated by the θb\theta^b. By the Frobenius theorem, this holds exactly when every point lies on an integral manifold of dimension n−sn - s.

Example: on ℝ3\R^3, the system generated by θ=dz\theta = dz has dθ=0d\theta = 0, so it is completely integrable. Its integral surfaces are the planes z=cz = c.

Example: on ℝ3\R^3, let θ=dz−y dx\theta = dz - y\, dx. Then dθ=dx∧dyd\theta = dx \wedge dy and θ∧dθ=dz∧dx∧dy≠0\theta \wedge d\theta = dz \wedge dx \wedge dy \ne 0, so the system is not completely integrable. It has integral curves, such as t↦(t,c,ct)t \mapsto (t, c, ct), but no integral surfaces.

Non-example: the exterior differential system on ℝ3\R^3 generated by the two-form dx∧dydx \wedge dy is not a Pfaffian system. It contains no nonzero one-form, so it is not generated by one-forms.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Differential form, concept, no evidence levelDifferential formconceptExterior differential system, concept, no evidence levelExterior differential…conceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTangent bundle, concept, no evidence levelTangent bundleconceptOne-form, concept, no evidence levelOne-formconceptFrobenius theorem, statement, no evidence levelFrobenius theoremstatementPfaffian system, definition, no evidence levelPfaffian systemdefinition
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  1. Differential form concept
  2. Exterior differential system concept
  3. Lie bracket of vector fields concept
  4. Smooth manifold concept
  5. Tangent bundle concept
  6. One-form concept
  7. Frobenius theorem statement
  8. Pfaffian system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Differential_ideal
About this page
TagDT60HNP41N
URIm2://DT60HNP41N
Kinddefinition
WritingW0 stub
MSC58A17, 58A15
DefinesPfaffian system
UsesExterior differential system, One-form, Differential form, Smooth manifold, Frobenius theorem

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